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Convergence and summable almost $$T$$-stability of the random Picard-Mann hybrid iterative process. (English) Zbl 1351.47051
Let $$(\Omega ,\xi ,\mu )$$ be a probability space, $$E$$ a Banach space, $$T:\Omega \times E\to E$$ a random operator, and $$x^*$$ the random fixed point of $$T$$. The authors study parametrized versions of the Mann and the Picard-Mann hybrid iterative processes for the approximation of $$x^*$$. They establish convergence and summable almost stability of these iteration schemes for the operator $$T$$ satisfying the following contraction-type condition: $\| x^*(\omega )-T(\omega ,y)\| \leq \theta (\omega )\| x^*(\omega )-y\|-\varphi (\| x^*(\omega )-y\| ,$ where $$\omega \in\Omega$$, $$y\in E$$, $$0\leq \theta (\omega )<1$$, $$\varphi$$ is nondecreasing and $$\varphi (t)>0$$ for $$t>0$$.
Reviewer’s remark: It is not difficult to see that a simpler version of the above condition, without any function $$\varphi$$, guarantees convergence and summable almost stability of the random Picard-Mann hybrid iteration scheme.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H40 Random nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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